课程： 6.5 多重不平等

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选择节多重不平等

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多重不平等
Do you know how to graph an inequality with more than one inequality symbol, like $\begin{aligned} - 4 < x < 5 ? \end{aligned}$
::您知道如何用 - 4 < x < 5 等多个不平等符号绘制不平等图吗 ?

Compound Inequalities
::多重不平等

C ompound inequalities are inequalities with more than one constraint on the possible values the solution can have.
::多重不平等是不平等,对解决办法可能具有的价值观有不止一种限制。

There are two types of :
::有两种类型:

Inequalities joined by the word “and,” where the solution is a set of values greater than a number and less than another number. These compound inequalities may appear in the form $\begin{aligned} x > a and x < b, \end{aligned}$ but are commonly are combined, like $\begin{aligned} a < x < b . \end{aligned}$ Possible values for $\begin{aligned} x \end{aligned}$ are ones that will make both inequalities true.
::在“和”一词中,“和”一词的解决方法是一组数值大于一个数值,少于另一个数值。这些复合不平等可能以 x>a 和 x<b 的形式出现,但通常是结合的,如<x<b。 x 的可能值是使不平等成为真实的值。

Inequalities joined by the word “or,” where the solution is a set of values greater than a number or less than another number. We write these inequalities in the form “ $\begin{aligned} x > a \end{aligned}$ or $\begin{aligned} x < b \end{aligned}$ .” Possible values for $\begin{aligned} x \end{aligned}$ are ones that will make at least one of the inequalities true.
::在“或”一词中,如果解决方案是一组数值大于或小于一个数字或小于另一个数字的数值,那么“或”一词加入不平等。我们以“x>a 或 x<b” 的形式写下这些不平等。 x 的可能值将至少使不平等成为事实。

You might wonder why the variable $\begin{aligned} x \end{aligned}$ has to be greater than one number and/or less than the other number; why can’t it be greater than both numbers, or less than both numbers? To see why, let’s take an example.
::您可能会想知道为什么变量x必须大于一个数字和(或)小于另一个数字; 为什么它不能大于两个数字, 或小于两个数字? 看看为什么, 让我们举一个例子。

Consider the compound inequality “ $\begin{aligned} x > 5 \end{aligned}$ and $\begin{aligned} x > 3 \end{aligned}$ .” Are there any numbers greater than 5 that are not greater than 3? No! Since 5 is greater than 3, everything greater than 5 is also greater than 3. If we say $\begin{aligned} x \end{aligned}$ is greater than both 5 and 3, that doesn’t tell us any more than if we just said $\begin{aligned} x \end{aligned}$ is greater than 5. So this compound inequality isn’t really compound; it’s equivalent to the simple inequality $\begin{aligned} x > 5 \end{aligned}$ . And that’s what would happen no matter which two numbers we used; saying that $\begin{aligned} x \end{aligned}$ is greater than both numbers is just the same as saying that $\begin{aligned} x \end{aligned}$ is greater than the bigger number, and saying that $\begin{aligned} x \end{aligned}$ is less than both numbers is just the same as saying that $\begin{aligned} x \end{aligned}$ is less than the smaller number.
::考虑“ x>5 和 x> 3 ” 的化合物不平等。 5个以上的数字是否大于5个,不大于3个? 否。由于5个大于3个,5个以上,所有大于5个以上的人都大于3个。如果我们说x大于5个和3个,那么没有比我们刚刚说x大于5个更能告诉我们更多。因此,这种化合物不平等并不真正复杂;这相当于简单的不平等x>5。这就是我们使用的两个数字都不会发生的情况;说x大于2个数字,与说x大于5个数字相同,说x少于2个数字,而说x少于2个数字与表示x少于较小数字相同。

Compound inequalities with “or” work much the same way. Every number that’s greater than 3 or greater than 5 is also just plain greater than 3, and every number that’s greater than 3 is certainly greater than 3 or greater than 5—so if we say “ $\begin{aligned} x > 5 \end{aligned}$ or $\begin{aligned} x > 3 \end{aligned}$ ,” that’s the same as saying just “ $\begin{aligned} x > 3 \end{aligned}$ .” Saying that $\begin{aligned} x \end{aligned}$ is greater than at least one of two numbers is just the same as saying that $\begin{aligned} x \end{aligned}$ is greater than the smaller number, and saying that $\begin{aligned} x \end{aligned}$ is less than at least one of two numbers is just the same as saying that $\begin{aligned} x \end{aligned}$ is less than the greater number.
::与“ 或 ” 相同的是“ 或 ” 。超过 3 或 5 的化合物不平等。每一个数字,如果大于 3 或 5 的,也只是 3 的,每个数字肯定大于 3 或 5 的,因此,如果我们说“ x > 5 或 x> 3 ” , 与仅仅说“ x > 3 ” 一样。说x 大于至少两个数字中的一个,与说x 大于小数字相同,说x 至少少于两个数字中的一个,与说x 少于更多数字相同。

Write and Graph Compound Inequalities on a Number Line
::数字行上的刻录和图形化合物不平等

Write the inequalities represented by the following number line graphs.
::写下以下数字行图表所代表的不平等。

a)
::a) (a)

The solution graph shows that the solution is any value between -40 and 60, including -40 but not 60.
::解决方案图显示,解决方案在 -40 至 60 之间具有任何值, 包括 -40 但不包括 60 。

Any value in the solution set satisfies both $\begin{aligned} x \geq - 40 \end{aligned}$ and $\begin{aligned} x < 60 \end{aligned}$ .
::解决方案集中的任何值均符合 x40 和x<60 。

This is usually written as $\begin{aligned} - 40 \leq x < 60 \end{aligned}$ .
::通常写为- 40x < 60 。

b)
:b) b)

The solution graph shows that the solution is any value greater than 1 (not including 1) or any value less than -2 (not including -2). You can see that there can be no values that can satisfy both these conditions at the same time. We write: $\begin{aligned} x > 1 \end{aligned}$ or $\begin{aligned} x < - 2 \end{aligned}$ .
::解决方案图显示,解决方案的任何值均大于1(不包括1)或小于-2(不包括-2)或小于-2(不包括-2)的值。您可以看到,没有值能够同时满足这两个条件。我们写道: x>1orx%2。

c)
:c) c)

The solution graph shows that the solution is any value greater than 4 (including 4) or any value less than -1 (including - 1). We write: $\begin{aligned} x \geq 4 \end{aligned}$ or $\begin{aligned} x \leq - 1 \end{aligned}$ .
::溶液图显示,溶液的任何值均大于4(包括4)或小于-1(包括-1)的任何值,我们写: x4orx1。

Graph the following compound inequalities on a number line.
::数字行的化合物不平等图如下。

a) $\begin{aligned} - 4 \leq x \leq 6 \end{aligned}$
::a) - 4x6

The solution is all numbers between -4 and 6, including both -4 and 6.
::解决办法是在-4和6之间的所有数字,包括-4和6。

b) $\begin{aligned} x < 0 \end{aligned}$ or $\begin{aligned} x > 2 \end{aligned}$
::b) x <0或 x>2

The solution is all numbers less than 0 or greater than 2, not including 0 or 2.
::溶液的所有数字小于0或大于2,不包括0或2。

c) $\begin{aligned} x \geq - 8 \end{aligned}$ or $\begin{aligned} x \leq - 20 \end{aligned}$
::c) x8 或 x20

The solution is all numbers greater than or equal to -8 or less than or equal to -20.
::溶液的数值均大于或等于-8或小于或等于-20。

Solve Compound Inequalities Using a Graphing Calculator (TI-83/84 family)
::使用图表计算计算器(TI-83/84家庭)

Graphing calculators can show you the solution to an inequality in the form of a graph. This can be especially useful when dealing with compound inequalities.
::图形计算器可以用图表的形式向您展示不平等的解决方案。在处理复合不平等时,这特别有用。

a) $\begin{aligned} 5 x + 2 (x - 3) \geq 2 \end{aligned}$
:a) 5x+2(x-3-3)%2

Press the [Y=] button and enter the inequality on the first line of the screen.
::按下 [Y=] 按钮,并在屏幕第一行输入不平等。

(To get the $\begin{aligned} \geq \end{aligned}$ symbol, press [TEST] [2nd] [MATH] and choose option 4.)
:获得符号, 按 [TEST] [第二 [MATH] 并选择选项 4 。 )

Then press the [GRAPH] button.
::然后按下[GRAPH]按钮。

Because the calculator uses the number 1 to mean “true” and 0 to mean “false,” you will see a step function with the $\begin{aligned} y - \end{aligned}$ value jumping from 0 to 1.
::因为计算器使用数字 1 表示“真实”, 0 表示“虚假”, 你会看到一个从 0 到 1 的 y- 值跳跃的阶梯函数。

The solution set is the values of $\begin{aligned} x \end{aligned}$ for which the graph shows $\begin{aligned} y = 1 \end{aligned}$ —in other words, the set of $\begin{aligned} x - \end{aligned}$ values that make the inequality true.
::解决方案集为 x 的值,图中显示 y=1 , 换句话说, 是使不平等成为真实的 x- 值组。

Note: You may need to press the [WINDOW] key or the [ZOOM] key to adjust the window to see the full graph.
::注意: 您可能需要按 [WINDOW] 键或 [ZOOM] 键来调整窗口以查看整张图。

The solution is $\begin{aligned} x > \frac{8}{7} \end{aligned}$ , which is why you can see the $\begin{aligned} y - \end{aligned}$ value changing from 0 to 1 at about 1.14.
::溶液为 x>87, 这就是为什么您可以看到 Y - 值在大约1. 14时从 0 到 1 变化。

b) $\begin{aligned} 7 x - 2 < 10 x + 1 < 9 x + 5 \end{aligned}$
::b) 7x-2 < 10x+1 < 9x+5

This is a compound inequality: $\begin{aligned} 7 x - 2 < 10 x + 1 \end{aligned}$ and $\begin{aligned} 10 x + 1 < 9 x + 5 \end{aligned}$ . You enter it like this:
::这是复合不平等: 7x-2 < 10x+1 和 10x+1 < 9x+5。您可以像这样输入 :

(To find the [AND] symbol, press [TEST] , choose [LOGIC] on the top row and choose option 1.)
:找到[AND]符号,按[TEST],在顶行选择[LOGIC],并选择备选案文1。 )

The resulting graph should look like this:
::结果的图表应该像这个样子 :

The solution are the values of $\begin{aligned} x \end{aligned}$ for which $\begin{aligned} y = 1 \end{aligned}$ ; in this case that would be $\begin{aligned} - 1 < x < 4 \end{aligned}$ .
::溶液是y=1的 x 值;在此情况下为-1 < x < 4。

c) $\begin{aligned} 3 x + 2 \leq 10 \end{aligned}$ or $\begin{aligned} 3 x + 2 \geq 15 \end{aligned}$
:c) 3x+2%10或3x+2%15

This is another compound inequality.
::这是另一个复杂的不平等。

(To enter the [OR] symbol, press [TEST] , choose [LOGIC] on the top row and choose option 2.)
:输入 [OR] 符号,按 [TEST],在顶行选择 [LOGIC],并选择选项2。 )

The resulting graph should look like this:
::结果的图表应该像这个样子 :

The solution are the values of $\begin{aligned} x \end{aligned}$ for which $\begin{aligned} y = 1 \end{aligned}$ --in this case, $\begin{aligned} x \leq 2.7 \end{aligned}$ or $\begin{aligned} x \geq 4.3 \end{aligned}$ .
::溶液是 x 的值,在此情况下是 y=1-, x=2.7 或 x4.3。

Examples
::实例

Example 1
::例1

Write the inequality represented by the following number line graph.
::写下以下数字行图代表的不平等。

The solution graph shows that the solution is any value that is both less than 25 (not including 25) and greater than -25 (not including -25). Any value in the solution set satisfies both $\begin{aligned} x > - 25 \end{aligned}$ and $\begin{aligned} x < 25. \end{aligned}$
::解决方案图显示,解决方案的任何值均小于25(不包括25)和大于25(不包括25)。设定的解决方案中的任何值均符合 x25andx<25。

This is usually written as $\begin{aligned} - 25 < x < 25. \end{aligned}$
::通常写为- 25 < x < 25 。

Example 2
::例2

Graph the following compound inequality on a number line.
::数字行的化合物不平等图如下。

$\begin{aligned} - 15 < x \leq 85 \end{aligned}$
::-15<x85

The solution is all numbers between -15 and 85, not including -15 but including 85.
::解决办法是15至85之间的所有数字,不包括15至85,但包括85。

Review
::回顾

Write the compound inequalities represented by the following graphs.
::撰写下图所示的复合不平等。

Graph the following compound inequalities on a number line.
::数字行的化合物不平等图如下。

$\begin{aligned} - 2 \leq x \leq 20 \end{aligned}$
::-2x 20 -2x 20

$\begin{aligned} x < 7 \end{aligned}$ or $\begin{aligned} x > 25 \end{aligned}$
::x7 或 x > 25

$\begin{aligned} x \geq - 100 \end{aligned}$ or $\begin{aligned} x \leq - 50 \end{aligned}$
::x100 或 x50

$\begin{aligned} - 1 < x < 200 \end{aligned}$
::-1 <x<200

$\begin{aligned} 2000 < x \leq 2001 \end{aligned}$
::2000<x_2001

$\begin{aligned} x \leq 1.56 \end{aligned}$ or $\begin{aligned} x > 1.78 \end{aligned}$
::x1.56 或 x> 1.78

$\begin{aligned} x > 0.0005 \end{aligned}$ or $\begin{aligned} x \leq - 0.03 \end{aligned}$
::x>0.0005 或 x 0.03

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

常规

多重不平等

Compound Inequalities ::多重不平等

Graph the following compound inequalities on a number line. ::数字行的化合物不平等图如下。

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Compound Inequalities
::多重不平等

Graph the following compound inequalities on a number line.
::数字行的化合物不平等图如下。

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)